KSGNS type construction for $\alpha$-completely positive maps on Krein $C^{\ast }$-modules

Abstract

In this paper, we investigate $\Phi$-maps associated to a certain type of $\alpha$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $\alpha$-completely positive maps on Krein $C^{\ast }$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $\alpha$-completely positive map and study the structure of minimal covariant KSGNS constructions.

20 pages, to appear in Complex Analysis Operator Theory

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